天山-6-4

If n is the product of the integers from 1 to 20 inclusive, what is the greatest integer k for which 2^k is a factor of n?
A. 10
B. 12
C. 15
D. 18
E. 20
Answer

请问解题的思路是怎样的?

百思不得其解啊。

对不起，用“天山 Q4”或者干脆“Q4”都没有搜到。

没有办法了才烦扰各位。

谢谢了！

n = 1 * 2 * 3 * 4 * 5... * 19 * 20

= (2 * 4 * 6 * 8 * 10... * 20) * (1 * 3 * 5 * 7...* 19)

= 2^10 * (1 * 2 * 3 * 4 * 5... * 10) * (1 * 3 * 5 * 7 ...* 19)

= 2^10 * (2 * 4 * 6 * 8 * 10) * (1 * 3 * 5 * 7 * 9) * (1 * 3 * 5 * 7... * 19)

= 2^10 * 2^5 * (1 * 2 * 3 * 4 * 5) * ...

= 2^15 * 2^3 * (1 * 3 * 5) *...

= 2^18 * ...